An experimental and theoretical study of electronic excitation and charge transfer processes in collisions between Cs(62S1/2) atoms and Na+(1S0) ions in the 0.30-4.00 keV energy range

Article abstract of DOI:10.1016/S0301-0104(02)00595-5

The collisional system Cs+Na⁺ with both species in their ground electronic state, has been studied using the crossed molecular beams technique and measuring the resulting fluorescent decay. Emissions from electronically excited Cs(6²P_3/2) and Cs(7²P_1/2,3/2) atoms have been detected as well as those from excited Na(3²P_1/2,3/2) and Na(4²D_3/2,5/2) produced by electron transfer. Cross-section values in the 0.30-4.00 keV energy range have been measured for the four different collision channels as well as the branching ratio between the resolved J-states for Cs(7²P_1/2,3/2). A qualitative interpretation for some of the measured emission cross-sections as a function of the collision energy has been performed using ab initio one-electron calculations done for the (CsNa)⁺ quasimolecule. Hemiquantal dynamical calculations have been carried out at selected collision energies including radial and angular non-adiabatic couplings, comparing the calculated cross-sections with the experimental data.

Full text of DOI:10.1016/S0301-0104(02)00595-5

Chemical Physics 281 (2002) 33–47
www.elsevier.com/locate/chemphys
An experimental and theoretical study of electronic
excitation and charge transfer processes in collisions between
2
þ 1
Cs(6 S ) atoms and Na ðS Þions in the 0.30–4.00 keV
1=2
0
energy range
a
a
a
a
a
J.de Andr ꢀe s , M.Sabid oꢀ , M.E. Aricha , M.Albert ꢀı , J.M. Lucas ,
b
F.X. Gadea , A.Aguilar
a,
*
a
Departament de Qu ꢀı mica F ꢀı sica, CeRQT/Parc cient ꢀı fic de Barcelona, Universitat de Barcelona, C/Mart ꢀı i Franqu eꢁ s 1,
8028 Barcelona, Spain
0
b
Laboratoire de Physique Quantique, I.R.S.A.M.C., Universit eꢀ Paul Sabatier, 118, Route de Narbonne, 31062 Toulouse, France
Received 27 February 2002
Abstract
þ
The collisional system Cs + Na with both species in their ground electronic state, has been studied using the crossed
molecular beams technique and measuring the resulting fluorescent decay.Emissions from electronically excited
2
2
2
2
Cs(6 P3=2) and Cs(7 P1=2;3=2) atoms have been detected as well as those from excited Na(3 P1=2;3=2) and Na(4 D3=2;5=2
produced by electron transfer. Cross-section values in the 0.30–4.00 keV energy range have been measured for the four
)
2
different collision channels as well as the branching ratio between the resolved J-states for Cs(7 P1=2;3=2).A qualitative
interpretation for some of the measured emission cross-sections as a function of the collision energy has been performed
þ
using ab initio one-electron calculations done for the (CsNa) quasimolecule.Hemiquantal dynamical calculations have
been carried out at selected collision energies including radial and angular non-adiabatic couplings, comparing the
calculated cross-sections with the experimental data. Ó 2002 Elsevier Science B.V. All rights reserved.
1
. Introduction
neutral processes are those taking place in inter-
stellar clouds, planetary ionospheres, gas discharge
and plasma deposition of thin metal films.Thus,
ion–atom collisions are a current research topic in
the field of Chemical Dynamics using molecular
beams techniques.Their interest becomes even
greater when both colliding particles are alkali
atoms, since it is easy to generate the corre-
sponding atomic beams and to measure the
decaying fluorescence emission from electronically
excited atoms, and also because their main
Collisions between ions and neutrals (atoms or
molecules) play an important role in different areas
of the Physics and Chemistry of low density plas-
mas and ionised gases.Relevant examples of ion–
*
Corresponding author.Te:l. +34-934021224; fax: +34-
34021231.
E-mail address: antonio@qf.ub.es (A.Aguilar).
9
0
301-0104/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved.
PII: S0301-0104(02)00595-5

34
J. de Andr ꢀe s et al. / Chemical Physics 281 (2002) 33–47
dynamical behaviour can be studied performing
relatively simple model calculations.Since the pi-
oneering experimental work on alkali ion–alkali
atom and alkali ion–atom systems done by Perel
et al.[1–3], Aquilanti et al.[4–10], Tolk et al.[11],
Andersen et al.[12], Østagaard et al.[13] and
Brunetti et al.[14], covering many symmetric and
asymmetric alkali ion–atom pairs, more elabo-
rated experiments involving alkali atoms excited
by polarised lasers to observe the specific align-
ment effects on electron transfer collisions have
been done by B €a hring et al.[15] and more recently
by Thomsen et al.[16].
Taking into account the electronic structure of
the alkali atoms it is not surprising that only the
single ns valence electron becomes excited in the
energy range currently under study.Then, results
on theoretical calculations can be obtained using
relatively simple models which take into account
the intrinsic non-adiabatic nature of the electronic
excitation or electron transfer processes.Aquilanti
hemiquantal dynamics are given in Section 5 and
conclusions are summarised in Section 6.
2. Experimental setup
As a detailed description of the experimental
crossed beam setup has been given already in
[22,29], here we will limit to briefly describe its
most relevant features.In this apparatus a beam of
þ
alkali ions (Na in the present case) is generated
by a thermionic source and accelerated by a 0.10–
4.00 keV electric field, focused by an einzel lens
system and later collimated.The ion beam crosses
2
at right angle the thermal neutral beam of CsðSÞ
atoms generated by the displacement reaction in
solid phase BaðsÞþ2CsClðsÞ!BaCl ðsÞþ2Cs
2
ðgÞ, which takes place in a heated oven.The
fluorescence emitted by decay of the excited elec-
tronic states produced in the collision is collected
by an optical system placed perpendicularly to the
collision plane at the scattering center, analysed by
a 50 cm monochromator and read by a photo-
multiplier.
[
17] improved on earlier studies of Perel et al.[2],
Melius et al.[18] and McMillan [19].Shingal et al.
20] performed semiclassical studies while more
[
recently Nakamura et al.[21] reported on a com-
plete picture of the two-state curve crossing
problems which can be applied to elastic and in-
elastic scattering as well as to perturbed bound
state problems.
A systematic study of collisions involving alkali
ions and alkali atoms [22–29] was done in our re-
search group from both experimental and theo-
retical points of view.One-electron ab initio
calculations using non-empirical relativistic
pseudopotentials to model the core electrons have
been successfully applied to several alkali ion–
atom pairs allowing an interpretation of the most
relevant experimental results.In this paper we re-
Background pressure is usually maintained in
À6
À7
the range 10 –10
mbar during experiments.
Recently, the experimental setup has been im-
proved by installing a computerised data-collec-
tion system which both shows on the screen the
fluorescence spectrum and accumulates the aver-
age fluorescence intensity measured along a given
time interval.
In order to report absolute cross-section values,
À
experiments on the Cs þe system were done to
calibrate our experimental system [30].To this end,
2
cross-sections for the inelastic process Csð6 S1=2Þþ
À
2
À
e !Csð6 P1=2;3=2Þþe were measured in arbi-
trary units and then compared with previously
reported data [31] in absolute units.
þ
port on the results obtained for the Cs þNa
collisional system.The paper is organised as fol-
lows: a brief description of the experimental ap-
paratus is contained in Section 2, while Section 3 is
devoted to present experimental results.A discus-
sion of the obtained data in terms of simple dy-
namical models as well as on our recent ab initio
3. Experimental results and discussion
3.1. Total and state-to-state cross-sections
þ
calculations performed on the ðCsNaÞ quasi-
When cesium atoms in their ground electronic
state collide with sodium ions, different excited
electronic states of the target Cs atom can be
molecule is presented in Section 4.Dynamical
calculations performed in the framework of the

J. de Andr ꢀe s et al. / Chemical Physics 281 (2002) 33–47
35
populated by direct excitation as well as those of
sodium atoms by an electron transfer processes.
Emissions for the following decay processes have
been observed and their total emission cross-sec-
tion determined at different collision energies:
These both lie in the infrared zone of the spectrum,
where background noise from our beam sources is
highest, which makes them difficult to characterise
in our experimental setup.Consequently, the
measurement of the cross-section has been pos-
2
2
þ
1
sible only for the more energetic Csð6 S1=2
Þ
ðaÞ Csð6 S1=2ÞþNa ðS
0
Þ
2
Csð6 P Þtransition.
3=2
2
þ
1
!
Csð6 P1=2;3=2ÞþNa ðS
Csð6 P3=2Þ!Csð6 S1=2Þ; k ¼8521:1 A;
0
Þ
2
Total emission cross-sections for the Csð6 S
Þ
1
=2
þ
1
2
þ
1
2
2
ꢂ
þNa ðS Þ!Csð6 P ÞþNa ðS Þprocess as a
0
3=2
0
function of the collision energy in the laboratory
frame (LF) in the 0.70–2.50 keV energy range are
given in Fig.1 together with the error bars re-
flecting experimental uncertainty.The low signal-
to-noise ratio limited the observable threshold
energy down to 0.70 keV in the LF.
2
þ
1
ðbÞ Csð6 S1=2ÞþNa ðS
0
Þ
2
þ
1
!
Csð7 P1=2;3=2ÞþNa ðS
Csð7 P1=2;3=2Þ!Csð6 S1=2Þ;
k ¼4593:2; 4555:4 A;
0
Þ
2
2
ꢂ
Since no other electronic excitations leading to
2
excited states of Cs decaying to the 6 P3=2 level
have been observed, cascade effects can be ne-
glected.LF emission cross-section values obtained
for the excitation process under consideration lie
in the same range as those previously found for the
2
þ
1
ðcÞ Csð6 S1 ÞþNa ðS Þ
=2
0
þ
1
2
!2
Cs ð6 S
0
ÞþNað3 P1=2;3=2
Nað3 P1=2;3=2Þ!Nað3 S1=2Þ;
k ¼5895:9; 5889:9 Aꢂ ;
Þ
2
þ
þ
systems Na–Na and Rb–Na in our group
þ
[
22,25], and for the K–Na system measured by
Aquilanti et al.[8].Cross-section values at 1000 eV
in the LF in the serie K, Rb, Cs against Na are
2
þ
1
ðdÞ Csð6 S1 ÞþNa ðS Þ
=2
0
þ
1
2
þ
!2
Nað4 D3
Cs ð6 S ÞþNað4 D3=2;5=2Þ
0
2
Þ!Nað3 P1=2;3=2Þ;
=2;5=2
k ¼5688:2; 5682:7 Aꢂ :
The electronic state of the excited alkali atom can
be characterised by the total electronic angular
momentum J and decay emissions to lower energy
0
states involving different J values are possible.In
this paper we report state-to-state cross-sections
0
rðJ
JÞfor the excitation processes (b) while
only total emission cross-sections (summed over
0
all possible J and J values) are reported for pro-
cesses (a), (c) and (d).
2
3
.2. Target excitation processes: Csð6 S1=2Þþ
þ
1
2
þ
1
2
Na ðS
Na ðS
0
0
Þ!Csð6 P1=2;3=2ÞþNa ðS
0
Þ; Csð7 P1=2;3=2Þþ
þ
1
Þ
The collision channel involving electronic exci-
2
tation of the target atom Cs(6 S1=2) to the first
excited states in collisions with sodium ions pro-
2
2
duces caesium atoms in the levels 6 P1=2 and 6 P3=2
Fig.1.Emission cross-sections versus LF (
Elab) and CMF
2
which decay to the ground level 6 S1=2 by fluores-
ꢂ
cent emission at 8943.5 and 8521.1 A, respectively.
2
(E
CM) collision energies for the formation of Cs(6 P3=2) in col-
2
þ
1
lisions between Cs(6 S ) and Na ðS Þ.
1=2
0

36
J. de Andr ꢀe s et al. / Chemical Physics 281 (2002) 33–47
À17
À16
À16
2
2
6
:5 Â10 ; 5:5 Â10
and 6:2 Â10
cm , re-
the state-to-state branching ratio emission cross-
0
0
spectively, where for the last one the 6 P1=2 con-
tribution cannot be included.According to these
results we can conclude that translational energy
in the LF becomes more effective in promoting
direct electronic excitation of target atoms to their
first excited level as their mass increase (the
sections (rðJ ¼1=2 J ¼3=2Þ=rðJ ¼1=2 J ¼
0
1=2Þ, where J and J stand, respectively, for the
final and initial atomic total electronic angular
momentum involved in the transition) is given in
Fig.3.As it can be seen in Fig.3 the branching
ratio is strongly energy dependent, varying from
about 0.4 at low collision energies to values be-
tween 1.5 and 1.8 at the highest ones. All these
values are lower than the statistical result of 2,
with the collision exit channel producing
þ
Na–Na system, for which direct excitation and
electron transfer processes are resonant has not
been included).In the simplest way this effect can
be justified taking into account that available
collisional energy in the center of mass frame
2
Cs(7 P3=2) being favoured when collision energy
2
(
CMF) increases from K to Cs, thus increasing the
increases, while that leading to Cs(7 P1=2) is es-
sentially energy-independent as it has been
process probability in the range of energies con-
sidered.
Another target excitation process also detected
0
found from the measured rðJ ¼1=2
J ¼1=2Þ
state-to-state cross-section [30].Moreover, in Fig.
3 we can appreciate some kind of structure,
particularly in the high collision energy range,
which could be attributed to interference effects,
but the uncertainty of the measured state-to-state
cross-sections does not allow an interpretation of
these oscillations.A similar branching ratio
dependence, with more pronounced oscillations,
þ
under our experimental conditions for the Cs–Na
2
system, leads to excited Cs(7 P1=2;3=2).In this case it
is possible to characterise fluorescent emissions at
ꢂ
2
2
4
4
593.2 A (from Csð6 S1=2
Þ
Csð7 P1=2Þ) and at
ꢂ
2
2
555.4 A (from Csð6 S1=2
Þ
Csð7 P3=2Þ) sepa-
rately, allowing state-to-state cross-sections to be
determined.In Fig.2 total emission cross-sections
2
þ
for Cs(7 P1=2;3=2) formation as a function of the
collision energy in the LF are plotted in the 0.50–
was found in the Rb–Cs [27] collision system but
for the electron transfer channel leading to
2
3
.00 keV energy range. LF energy dependence of
Cs(6 P1=2;3=2).
Fig.3 .C ross-section branching ratio between processes leading
2
2
2
Fig.2 .T otal emission cross-sections versus LF (
E
lab) and CMF
to Cs(7 P3=2) and Cs(7 P1=2) in collisions between Cs(6 S1=2) and
2
þ
1
(E
CM) collision energies for the formation of Cs(7 P1=2;3=2) in
Na ðS
0
Þas a function of the LF (Elab) and CMF (ECM) colli-
2
þ
1
collisions between Cs(6 S1=2) and Na ðS Þ.
0
sion energies.

J. de Andr ꢀe s et al. / Chemical Physics 281 (2002) 33–47
37
2
3
.3. Electron transfer processes: Csð6 S1=2Þþ
transfer can only be achieved by performing full
dynamical calculations including the manifold of
electronic potential energy curves correlating with
the asymptotic atomic states and taking into ac-
count all non-adiabatic couplings between them.
Nevertheless, to give a qualitative interpretation of
the experimental results, simple models can also be
useful.To this end, experimental data have been
fitted to the linear two-state Landau–Zener–Stuc-
kelberg (LZS) model [32].In Fig.4 continuous
lines represent the best fitting of the experimental
excitation function to this model for a rotationally
coupled R–P transition.It can be seen that this
model can reproduce well the experimental data,
but a proper fit of the cross-section values for the
whole energy range requires two different sets of
parameters for the corresponding LZS equation.
According to this simple model, the elec-
þ
1
þ
1
2
þ
1
Na ðS
0
Þ!Cs ð6 S
0
ÞþNað3 P1=2;3=2Þ; Cs ð6 S Þþ
0
2
Nað4 D3=2;5=2
Þ
Total emission cross-sections (where contribu-
tions from J ¼1=2 and J ¼3=2 levels of excited
Na atoms are considered together) as a function of
the collision energy in the LF for the exit channel
producing Na(3 P1=2;3=2) in the 0.30–3.50 keV en-
ergy range are shown in Fig.4.This is the most
2
þ
important channel in Cs + Na collisions.For it,
absolute cross-section values are about two orders
of magnitude higher than those for the funda-
2
2
mental electronic transition (6 S1
6 P1=2;3=2)
=2
for Cs atoms and also somewhat higher than the
emission cross-section values obtained previously
for equivalent electron transfer process in the
2
þ
1
Rbð5 S1=2ÞþNa ðS Þcollision system [25].This
0
2
þ
1
fact can be qualitatively interpreted considering
that Cs is more readily polarised than Rb (when
both are in the ground state), which facilitates the
tron transfer process Csð6 S Þþ Na ðS Þ!
1
=2
0
þ
1
2
Cs ð6 S ÞþNað3 P
0
1=2;3=2
Þ can be easily inter-
preted as being controlled by a rotational non-
adiabatic coupling between an entrance channel of
1
target valence ns electron transfer to the imping-
þ
þ
ing ion Na .
A quantitative interpretation of the experi-
R
symmetry, correlating asymptotically with
þ
2
1
Csð6 S ÞþNa ðS Þ, and an exit one of P
1=2
0
1
1
mental data for the Cs(6s ) to Na(3p ) electron
symmetry and correlating with product
1
þ
2
Cs ð6 S
0
Þþ Nað3 P1=2;3=2Þ, with the asymptoti-
þ
cally degenerated R channel playing a minor role.
However, the limitations of the model are evi-
denced by its inability to describe adequately the
cross-section energy dependence in the whole en-
ergy range.This was to be expected considering
the manifold of electronic states involved along the
collision.In fact, any charge transfer or target
excitation along the collisional event produces a
mixture of the different electronic states that can
be associated with the quasimolecule formed by
the collisional partners.
The second electron transfer process mea-
þ
2
þ
1
1
sured is Csð6 S ÞþNa ðS Þ!Cs ð6 S ÞþNa
1=2
0
0
2
ð4 D
Þ where the produced excited atoms
5
=2;3=2
2
Nað4 D3=2;5=2Þ decay to the lower energy level
2
Na(3 P1=2;3=2).Its total emission cross-section as a
function of the LF collision energy in the range
0
.50–2.00 keV is given in Fig. 5. Values for this
2
Fig.4.Total emission cross-section for the electron transfer
2
2
Nað4 D3=2;5=2 transition have
been taken into account in the calculation of the
emission cross-section associated with the forma-
Þ
Nað3 P1=2;3=2
Þ
process leading to Na(3 P1=2;3=2) versus LF (Elab) and CMF
2
(
E
CM) collision energies in collisions between Cs(6 S1=2) and
þ
1
2
Na ðS
0
Þcorrected by the cascade effect from Na(4 D3=2;5=2
)
2
decay.
tion of Na(3 P1=2;3=2) to correct it for the cascade

38
J. de Andr ꢀe s et al. / Chemical Physics 281 (2002) 33–47
þ
and excited states of the ðCs–NaÞ system which
include all electrons are still too expensive.How-
ever, for systems like the present one we can take
advantage of the pseudo-monoelectronic character
of the quasimolecule formed along the collision
using pseudopotential techniques [33], which we
have already applied to other systems [25,27,29]
providing a good qualitative interpretation of the
measured data.In the present case, non-empirical
pseudopotentials derived by Durand and Barthelat
[
34] have been used to replace sodium (1s2s2p) and
caesium (1s2s2p3s3p4s3d4p5s4d5p) electron cores,
including also relativistic parameters for the latter
[
35].Then, adiabatic potential energy curves have
been calculated using a restricted open-shell Har-
tree–Fock (ROHF) technique.Because of the high
polarizability of the atoms involved, the calcula-
tions were performed considering the core–valence
correlation by including a core polarisation po-
tential (CPP) in the Fock operator using a for-
malism proposed by Foucrault et al.[36].
Since the (Cs–Na) system involves large cores
and in our collision energy range the colliding
particles can approach each other within less than
Fig.5.Total emission cross-section for the electron transfer
2
process leading to Na(4 D3=2;5=2) as function of the LF (Elab
)
and CMF (ECM) collision energies in the Cs þNaþ collisional
system.
effect.Absolute cross-section values for this pro-
cess are about one order of magnitude lower than
þ
2
those measured for the Na(3 P1=2;3=2) formation
and about one order higher than emission cross-
sections for the equivalent electron transfer pro-
ꢂ
1 A, the core–core repulsion has not been calcu-
þ
cess in the Rb + Na system.While the fitting is
not so good as for the previously considered
lated as a pure coulombic value, but as the energy
þ
2þ
þ
difference between the CsNa and Na and Cs
3
function can be described by the LZS model (not
p
3s transition, the experimental excitation
in an all-electron calculation with a minimal basis
set [37,38].The single valence electron has been
described by extended Gaussian type orbitals
(GTO) basis sets: [7s, 6p, 5d, 2f/5s, 5p, 4d, 2f] for
Na [39] and [7s, 6p, 6d, 2f/6s, 6p, 5d, 2f ] for Cs
[40].Energy calculations were performed at inter-
nuclear distances (R) ranging from 1.00 to 50.00
a.u.
shown in Fig.5) considering a R–R non-adiabatic
þ
coupling between the R entrance channel corre-
lating asymptotically with colliding particles and
þ
the R component exit channel correlating with
þ
1
2
collision products Cs ð6 S
0
ÞþNað4 D3=2;5=2Þ.
Using this GTO basis set, 37 adiabatic potential
energy curves corresponding to 56 molecular states
have been calculated with high reliability.These
curves correlate asymptotically with atomic states,
whose energy diverge from the corresponding ex-
perimental values by 1% or less.According to their
symmetry the 37 adiabatic curves considered are
þ
4. Ab initio calculations for the (Cs–Na) collisional
system
In order to improve our understanding at
atomic level of the mechanism leading to excited
states of both Cs and Na atoms, we have to get
information on all the potential energy curves de-
scribing interactions between the collision partners
as well as on the non-adiabatic couplings between
them.Even with the continuously increasing su-
percomputational facilities nowadays available,
high level and full ab initio calculations for ground
þ
distributed as follows: 18R , 12P, 6D and a single
U, and they describe the asymptotic neutral atom–
þ
þ
ion pair (Cs þNa or Cs þNa) states in which
the Cs and Na atoms have electronic configura-
tions ranging from (of ordered increasing energies)
1
1
1
1
Cs(6s ) to Cs(7d ) and Na(3s ) to Na(5p ), re-

J. de Andr ꢀe s et al. / Chemical Physics 281 (2002) 33–47
39
spectively.For clarity reasons, only those calcu-
lated potential energy curves whose asymptotical
electronic configuration lies in the energy range
þ
1
bounded by the fundamental (Cs þNað3s Þ) and
the most excited one observed experimentally
(
Sometimes a qualitative interpretation of the
þ
1
Cs þNað4d Þ) are given in Fig.6.
experimental excitation functions can be given in
terms of the energy evolution of the adiabatic
molecular orbitals (MOs) as a function of R.This
is the case of the electron transfer process
2
þ
1
þ
1
2
Csð6 S1=2ÞþNa ðS
0
Þ!Cs ð6 S
0
ÞþNað3 P1=2;3=2
Þ
whose cross-section-energy dependence has been
discussed previously in terms of a rotationally
non-adiabatic LZS model.In Fig.7 the electronic
Fig.7 .E volution of the lowest adiabatic MOs as a function of
internuclear distance for the Cs þNaþcollisional system.
energy (without the core–core repulsion) evolution
is given as a function of R for the adiabatic MOs
correlating with the ground state (1r), the en-
trance channel (2r) and the exit ones (3r and 1p)
asymptotically degenerated.It can be seen how the
entrance channel 2r crosses the exit channel 1p
þ
1
(
correlating with Cs þNað3p Þ) at about 7.50 a.u.
After this, the 2r comes very close to the 1p at
about 2.50 a.u., and for lower R-values both MOs
evolve practically in parallel.The 3 r exit channel
also correlating with Cs þNað3p Þ) runs ener-
getically very separated from the entrance channel
2r), both MOs evolving also in a parallel way at
þ
1
(
(
very short (below 2.00 a.u.) internuclear separa-
tions.The 2 r–1p crossing justifies the rotational
non-adiabatic LZS fit at low and middle collision
energies, then allowing to interpret the experi-
mental data as dominated by the mentioned
crossing and in terms of a two states model.As
collision energy increases, atomic cores can ap-
proach closer and then non-adiabatic transitions
2r–1p can also take place at R ¼2:50 a.u. or less.
This behaviour can justify the good fit, at least
qualitatively, but the basic inadequacy of such a
two states model also becomes evident as ought to
be expected considering the manifold of potential
energy curves which can be explored along the
þ
Fig.6 .A diabatic potential energy curves for the (CsNa) ionic
system versus internuclear distances (R) obtained by ab initio
þ
calculations (see text).Continuous lines ( ———) stand for R
þ
states, dotted lines (ÁÁÁ) stand for P and dot-dash ones (-Á-Á-)
stand for D states.Only energy curves for asymptotical ion–
þ
atom states whose energy lies between the ground Cs þ
1
þ
1
Nað3s Þelectronic configuration and the excited Cs þNað4d Þ
one are shown.

40
J. de Andr ꢀe s et al. / Chemical Physics 281 (2002) 33–47
collision process and which play an important role
in the population of the atomic states.For in-
stance, let us consider the charge transfer process
work of hemiquantal mechanics [45] using a
computer code implemented by us in our compu-
tational facilities.In this method the collision
evolution of the electronic quantum degrees of
freedom is described by the time-dependent
Schr o€ dinger equation for the complete electronic
wave function W.This function is expanded in a
1
þ
leading to Nað4d ÞþCs whose excitation func-
tion can be acceptably fitted using a radial LZS
model.An analysis (not reported here) of the
evolution of the adiabatic MOs 14r, 8p and 4d
1
correlating with Na(4d ) and 2r for the entrance
set of the adiabatic molecular electronic wave
functions previously obtained W ¼
P
channel along R, shows no crossings between them
in all the range of internuclear separation.It is
then clear that any interpretation, even qualitative,
requires more elaborated models [21,41] or dy-
namical calculations.
c
i
ðtÞU
i
i
where complex coefficients c
i
ðtÞare function of
time t (or, equivalently, a function of R, provided
that R is a function of time).The remaining de-
grees of freedom, which describe nuclear motion
(in the CMF) in terms of the polar coordinates
ðR; HÞ, evolve according to the four differential
_
_
5
. Dynamical calculations
coupled hemiquantal equations for R; H; p_ and
R
p_
.For any particular collision, the set of hemi-
H
As has been stated before, in collisional systems
quantal equations describing the nuclear motion is
solved together with the electronic time-dependent
Schr o€ dinger equation, which involve both the ra-
dial and rotational non-adiabatic couplings [46].
In the hemiquantal approach the accuracy of the
results is limited by that of the potential energy
curves and non-adiabatic couplings, as well as by
the number of adiabatic molecular states used in
the adiabatic expansion of W.
In the range of the collision energies studied, the
orbital angular momentum (l) of the ion–atom
system can reach very high values.On the other
hand, changes in the projection of the total elec-
tronic orbital angular momentum are small and
the total electronic spin is conserved in any colli-
sion.Thus, the total angular momentum K of the
system is given by K ’l.In order to calculate
collision cross-sections, the coupled set of classical
and quantum degrees of freedom of the system are
solved for a fixed set of initial conditions: relative
þ
like the ðCsNaÞ and even at the lowest experi-
mental energies, we must consider all the electronic
states that can be populated along the collision.
Dynamical calculations require to know both the
evolution of the electronic states and all couplings
between them.In the framework of the adiabatic
representation [42] the Born–Oppenheimer elec-
tronic hamiltonian matrix is diagonal, the elec-
tronic excitation being induced by nuclear motions.
Then the corresponding non-adiabatic couplings
between the different adiabatic electronic states
must also be calculated.According to the ab initio
calculations reported in the previous section we
have obtained 56 electronic states (corresponding
to 37 adiabatic potential energy curves) asymptot-
ically very well described among a manifold of 127
electronic states.Non-adiabatic couplings for these
states have been calculated from the set of adia-
batic electronic wave functions fU gR-dependents.
i
For the present case of two colliding particles [43]
we need to calculate the so-called radial and rota-
tional couplings matrix elements given by
0
position (R ), initial electronic entrance channel
cinitðt Þ, collision orbital angular momentum (l) (or
0
the equivalent classical impact parameter) and
collision energy (E).All hemiquantal trajectories
^
i, respectively, where L^
hU
j
jo=oRjU
i
i; hU
j
jL
y
jU
i
y
is
the y-component of the orbital angular momentum
operator in the rotating frame.These matrix ele-
ments were evaluated numerically [44] using the
finite differences method on a grid of points along
R and interpolated using a cubic spline method.
start at R
negligible and the initial electronic state is the
0
¼50:00 a.u. where interactions are
þ
second R one correlating with the experimental
2
þ
1
Csð6 S1=2ÞþNa ðS Þentrance channel.At fixed
0
collision energies, cross-section values are ob-
tained until convergence on l.The c ðtÞcoefficients
þ
Dynamical calculations for the ðCsNaÞ col-
i
liding system have been performed in the frame-
involved in the complete electronic wave function

J. de Andr ꢀe s et al. / Chemical Physics 281 (2002) 33–47
41
expansion are l-dependent as is also the calculated
transition probability from the initial state (i) to
any possible exit channel (f) at the end of the
hemiquantal trajectory ðt !1Þ.Then total tran-
sition probability is given by
but these oscillations become both reduced in in-
tensity and smoothed for high l values.This be-
haviour could be explained considering that the
collision orbital angular momentum introduces a
centrifugal potential whose effect is to preclude
partially the participation of the excited states at
short distances, thus making a particular excita-
tion contribution less favourable.At fixed collision
energies (E) and for a given electronic transition,
the total transition probability (Eq.(1)) is a func-
tion only of the orbital angular momentum PðlÞ.
In Fig.8 are given (for a CMF collision energy of
425 eV) the opacity functions PðlÞÁð2l þ1Þfor the
l
Pf iðEÞ¼jc
f
ðl; 1Þj:
ð1Þ
Total inelastic excitation and/or electron transfer
cross-sections are given by the well-known equa-
tion
lmax
X
p
2
rf iðEÞ¼2
ð2l þ1Þjc ðl; 1Þj;
ð2Þ
f
lE
l¼0
1
1
Cs(6s Þexcitation to Csð7p Þcorrelating with both
þ
þ
2
2
where l is the reduced mass of ðCs–NaÞ and lmax
the maximum collision orbital angular momentum
leading to non-zero probabilities for the process
considered.
quasimolecular states 10 R and 6 P, and for the
1
2
þ
formation of Na(3p ) correlating with 3 R and 1
P and induced by collisions between Cs(6s ) and
2
þ
1
þ
0
Na ð3s Þ.Since these excited Cs and Na atoms can
be produced both by radial and rotational non-
adiabatic coupling, the contributions to the total
opacity function are shown separately.The figure
shows the oscillatory behaviour of the opacity
functions that was to be expected from the quan-
tum mechanical solutions of the quantum degrees
of freedom (the electronic subsystem).For these
transitions at the specified collision energies, the
opacity functions are fully converged for a maxi-
mum l value of 25.500 a.u. Both R –R opacity
functions show similar behaviour, their probabili-
ties being of the same order of magnitude and very
low for l < 5:000 a.u. In the case of the R –P
opacity functions for the charge transfer process
5
.1. Results and discussion
In order to obtain the most accurate results
possible with our computational facilities and
CPU time allowances, the adiabatic expansion has
2
þ
þ
been performed over the 18 lowest energy
2
R
states and the 12 lowest P ones of the ðCsNaÞ
system, the corresponding dynamical calculations
being carried out for eight collision energies in the
low range of experimental data.From a detailed
analysis of the collision time evolution of the
complete electronic wave function for a selected
number of hemiquantal calculations, some general
trends have been observed which are essentially
independent either of the collision orbital angular
momentum or of the energy.It can be appreciated
that non-adiabatic interactions appear at distances
R as large as 35–30 a.u., causing a mixture of
different excited R or P states.The time contri-
bution to these states oscillates with individual
values, usually reaching their maximum at R val-
ues ranging from 20 a.u. to their corresponding
turning point.In the second half-collision, when
both nuclei are receding apart, a similar oscillatory
contribution from each state also appears, the
corresponding coefficients becoming stabilised at
some R about 25–30 a.u. At a fixed collision en-
ergy, oscillatory behaviour for each electronic state
contribution is strong and sharp for low values of l
þ
þ
þ
1
leading to Na(3p ) only large impact parameters
(for l > 10:000 a.u.) are found to give an impor-
tant contribution, while for the Cs excitation
1
leading to Cs(7p ) all impact parameter values
contribute, but their probabilities are lower by one
order of magnitude or more that those involved in
the charge transfer discussed previously.Similar
qualitative behaviours have also been observed for
other collision energy values.
Total cross-sections for all different collision
channels involved in the adiabatic expansion of the
complete electronic wave function have been ob-
tained by hemiquantal calculations at different
CMF collision energies.In Table 1, it can be seen
the lmax values at the selected collision energies,
together with the total cross-sections obtained

42
J. de Andr ꢀe s et al. / Chemical Physics 281 (2002) 33–47
1
Fig.8.Opacity functions PðlÞÁð2l þ1Þ(see text) at 425 eV CMF collision energy for target excitation (to Cs(7p )) and electron
transfer (to Na(3p )) processes from reactants in their ground electronic state.Lower panel: R –R and R –P opacity function
1
þ
þ
þ
1
þ
2
þ
2
þ
þ
components leading to Csð7p ÞþNa correlating with both the 10 R and the 6 P quasimolecular states.Upper panel: R –R and
þ
þ
1
2
þ
2
R –P opacity function components leading to Cs þNað3p Þcorrelating with both the 3 R and the 1 P quasimolecular states.
Table 1
Hemiquantal calculated cross-sections and lmax values for the formation of different asymptotic atomic states of Cs and Na in collisions
between Cs and Na in their respective electronic ground states
þ
1
2
1
2
1
2
1
2
1
2
ECMF (eV)
l
max (a.u.)
rðCsð6p PÞÞ
rðCsð7p PÞÞ
rðNað3p PÞÞ
rðNað4d DÞÞ
rðNað3d DÞÞ
4
6
7
8
25
00
70
50
25,500
32,400
39,100
43,800
50,200
54,000
57,000
1.25
1.77
1.95
2.07
2.45
2.68
2.89
0.91
1.22
1.38
1.47
1.46
1.51
1.65
2.40
3.04
3.16
3.29
3.24
3.19
3.30
0.807
1.28
2.32
2.73
3.47
4.06
4.50
4.09
6.76
7.74
8.06
8.33
8.11
8.09
1
1
1
025
195
275
À15
2
Cross-section values have to be multiplied by 10
and given in cm .
from Eq.(2) for the experimentally observed exit
channels as well as for the channel leading to
We must proceed carefully before comparing
calculated and experimental cross-section values.
At a given collision energy from hemiquantal cal-
1
Na(3d D).
2

J. de Andr ꢀe s et al. / Chemical Physics 281 (2002) 33–47
43
culations, cross-section for the formation of a gi-
ven atom (Cs or Na) in a well-defined state is
obtained independently of the final measured
property which characterises this final atomic
state.Moreover, the number of possible final
states and the calculated transition probabilities
are limited by the adiabatic expansion done.Ex-
perimentally, at the same collision energy a large
number of both excited target Cs states and excited
Na ones (formed by electron transfer) are open,
and they have not been included in the adiabatic
expansion of the complete electronic wave func-
tion.In addition, some states can decay to lower
energy ones, producing either a cascade effect or
decaying to other excited states which are not
characterisable experimentally.Consequently,
some limitations can appear in the comparison of
calculated cross-section with experimental emis-
sion cross section data.
values being lower and consequently in better
agreement with measured data.
Calculated total excitation cross-sections for
1
2
Cs(7p
P1=2;3=2) are in average one order of mag-
nitude larger than the measured emission cross-
section values for decay to the ground atomic level
1
2
Cs(6s S1=2) as can be seen in Fig.9(b).This was
1
2
to be expected considering that Cs(7p P1=2;3=2
1
)
2
can also decay to Cs(7s
1
S
1=2) (and even from this
2
to the Cs(6p P1=2;3=2)), the calculated excitation
cross-section for Cs(7p
1
2
P1=2;3=2) being conse-
quently overestimated.From the available infor-
mation on the spectral lines for the Cs 7s 7p
decay [47] and the total measured intensity in the
laboratory for each transition, the total intensity
for the Cs 7s 7p (not observed in our experi-
ment) has been estimated [48], assuming similar
values for the transition dipolar moments in-
volved.Results thus obtained predict that the in-
tensity of the non observed Cs 7s 7p transition
should be about 0.9 times that for the Cs 6s 7p
transition.
In the case of calculated cross-section leading
1
2
to Cs(6p
P1=2;3=2), for which the spin–orbit
interaction has not been included, results can-
not be directly compared with experimental data
since only the J ¼3=2 component of the
Using this result we have obtained the corrected
1
2
excitation cross sections for Cs(7p P1=2;3=2) from
the original calculations and these values are also
included in Fig.9(b).It can be seen how the cor-
rected excitation cross-sections lie not far now
from the measured ones and are of the same order
of magnitude.This improved agreement seems to
point out that, although not measurable under our
experimental conditions, the Cs 7s 7p decay
plays an important role in diminishing the original
1
2
Cs(6p
P1=2;3=2) energy levels has been measured.
Calculated cross-section values can be expected to
be larger than measured data considering that
they include all the states for the spectral term
1
2
Cs(6p P) and this agrees quite well with the
data shown in Fig.9(a).In a first approximation,
and assuming that the cross-section branching
ratio for J ¼3=2 and J ¼1=2 level components is
statistical, excitation cross-section leading to Cs
1
2
population of the Cs(7p
P1=2;3=2) levels.
1
2
(
6p
P
3=2) can be obtained and is also repre-
In the same way as target excitation cross-sec-
tions, hemiquantal calculations also allow to study
charge transfer processes, in particular those
sented in Fig.9(a) where it can be seen that, al-
though the corresponding absolute values are
close to the experimental ones they lay outside
the error bars.A possible explanation of the
discrepancy between experimental and estimated
values can be that branching ratios could be en-
ergy-dependent (as it has been found in the pre-
sent paper for the Cs 6s 7p transition and in
other systems [27–29]).If the branching ratio
energy dependence for the Cs 6s 6p is similar
to that measured for Cs 6s 7p, in the low col-
lision energy range (as is the case in the present
1
2
1
2
leading to Na(3s
P
1=2;3=2) and to Na(4d
3=2;5=2
D )
þ
which have also been measured for the ðCsNaÞ
collision system.Calculated cross-section values
for the first excited levels of Na are shown in Fig.
9(c), together with the experimental emission
1
2
cross-section for Na(3s
P1=2;3=2).There it can be
seen how calculated values are lower than the ex-
perimental ones.Moreover the shape of the cross
section vs.energy curves is not quite the same in
the range of energies considered.The values for
emission cross-section seem to point out that cas-
cade effects from higher excited states do contrib-
1
2
calculation), the cross-sections for Cs(6p P3=2
reported in Fig.9(a) would be overestimated, real
)

44
J. de Andr ꢀe s et al. / Chemical Physics 281 (2002) 33–47
2
þ
1
Fig.9.Experimental and computed emission cross-sections for the different observed exit channel in the Cs ð6 S1=2ÞþNa ðS
0
Þ
1
2
collision system as a function of the collision energy (Elab and ECM) (see text).(a) Cs(6p 3=2) formation: ( ) experimental, (N)
P
ꢀ
1
2
1 2
calculated, (r) estimated; (b) Cs(7p
P
1=2;3=2) formation: (
ꢀ
) experimental, (r) corrected; (c) Na(3p
1
2
P
1=2;3=2) formation: ( )
ꢀ
1
1
experimental, (N) calculated, (r) calculated Nað3p ÞþNað3d Þ; (d) Na(4d
D3=2;5=2) formation: (
ꢀ
) experimental, (N) calculated.
1
2
ute to Na(3s P1=2;3=2) population.In fact, exper-
imental data reported in Fig.4 have been corrected
the necessary correction of the experimental exci-
tation function cannot be carried out for lack of
data on the Na 3p 3d transition.However, as-
1
2
for the cascade effect from Na(4d
but corrected values lie still within the error bars.
On the other hand, decay from Na(3d D3=2;5=2),
which cannot be experimentally measured (its
ꢂ
emission is at about 8194 A, in a high-noise zone)
could also be responsible for this discrepancy.If
1
the lifetime of excited Na(3d ) is shorter than that
D3=2;5=2) decay,
1
suming that Na(3d ) lifetime is much shorter than
1
2
that for Na(3p ), then the emission cross-section
1
2
from Na(3p
P1=2;3=2) can be approximately eval-
uated by adding both calculated cross-sections
1
1
leading to Na(3p ) and to Na(3d ).As can be
appreciate in Fig.9(c), hemiquantal emission
cross-sections values calculated using this ap-
proximation are in quite good agreement with
1
of Na(3p ), then the true measured emission cross-
1
section for Na(3p
2
P1=2;3=2) would be lower, but

J. de Andr ꢀe s et al. / Chemical Physics 281 (2002) 33–47
45
Table 2
Hemiquantal calculated cross-sections for the elastic
experimental data considering the experimental
error in the range of energies calculated.
The last experimental emission cross-section
1
2
þ
1
2
þ
Csð6s SÞþNa and dark Nað3s SÞþCs exit channels in
þ
collisions between Cs and Na in their respective electronic
ground states
þ
measured in the ðCsNaÞ collision system is that of
1
2
Na(4d
this case, apart from the lack of experimental in-
D3=2;5=2) produced by electron transfer.In
1
2
1 2
ECMF (eV)
rðCsð6s SÞÞ
rðNað3s SÞÞ
1
formation, although the excited Na(4f ) can decay
425
2.90
3.10
3.60
4.30
4.60
4.30
4.50
0.30
0.20
0.17
0.16
0.13
0.12
0.12
6
7
8
00
70
50
1
to Na(4d ), since the corresponding energy levels
for both electronic configurations are almost de-
1
generate, the 4d configuration can be expected to
1025
1
1
195
275
have a very long lifetime, when compared with this
decay, thus exerting a negligible influence on the
1
population of levels associated with the 4d con-
À14
Cross-section values have to be multiplied by 10
2
and
given in cm .
figuration.In Fig.9(d) calculated hemiquantal
cross-sections are reported together with measured
emission cross-section and it can be seen that
agreement between both data sets is good enough
considering the experimental error.
and cross-section for the electron transfer process
leading to Na atom in the electronic ground state
1
2
þ
Nað3s SÞþCs (the so-called ‘‘dark’’ channel)
Dynamical calculations also permit to obtain
þ
for which values are given in Table 2.Total elastic
þ
1
2
þ
the contribution of the radial ðR –R Þand angular
(Csð6s SÞþNa ) cross-sections have the highest
values by about one order of magnitude for all
energies studied, which is in agreement with the
predominance of elastic processes in elementary
inelastic collisions.For the dark exit channel
þ
ðR –PÞnon-adiabatic couplings to the total cal-
þ
culated cross-section values in the basis set of R
and P states used.As the experimental entrance
þ
channel correlates with a molecular state of R
type, only the radial coupling contributes to cross-
1
2
þ
producing Nað3s SÞþCs , cross section values
show an opposite energy dependence in compari-
son with other excitation or charge transfer pro-
cesses measured and of the same order of
magnitude in the range of energies considered.
sections for channels leading to Cs or Na atoms
1
with a ns electronic configuration, while rota-
tional coupling can become important when other
1
atomic electronic configurations like np are
possible.A case in point is the formation of
1
2
1 2
Cs(6p P) and Cs(7p P), where both radial and
angular coupling contributions to the total cross-
section are 12.8% and 87.2% and 53.0% and
6. Concluding remarks
4
7.05%, respectively, at 425 eV in the CMF. For
Electronic excitation of Cs target atoms
and charge transfer of impinging sodium atoms
in the (CsNa) collision system in the energy
range of 0.30–4.00 keV have been measured in
crossed molecular beams experiments.Emission
1
2
the electron transfer process leading to Na(3p P)
the radial contribution is 42.5% while the angular
one is 57.5%. Both radial and angular couplings
change with collision energy but the variation do
not follow any particular trend.
þ
1
2
1 2
cross-section for Cs(6p
1
P
and Na(4d
3=2), Cs(7p
P1=2;3=2),
2
1
2
Among the 18 asymptotic atomic electronic
configurations for Cs and Na atoms included in
the complete electronic wave function of the col-
liding system, for all collision energies studied the
Na(3p
P
1=2;3=2
)
levels have been obtained measuring fluores-
cence decay.In the case of Cs(7p
3=2;5=2
D ) atomic
1
2
P1=2;3=2) levels a
non-statistical energy dependent cross section
branching ratio for J ¼3=2 and J ¼1=2 has been
measured, with the population of J ¼3=2 level
increasing at the highest energies values until it
seems to tend to the statistical value.Cascade ef-
most favoured exit channel is Nað3d DÞþCsþ
1
2
whose cross-section values average about double
1
2
than those for Na ð3p PÞformation.
Other results obtained from hemiquantal dy-
namical calculations are total elastic cross-sections
1
2
fects on the Na(3p
P1=2;3=2) cross-sections from

46
J. de Andr ꢀe s et al. / Chemical Physics 281 (2002) 33–47
1
2
[
2] J. Perel, H.L. Daley, J.M. Peek, T.A. Green, Phys. Rev. 23
(
Na(4d
their influence is negligible.
D3=2;5=2) decay have been considered, but
1969) 677.
[
[
3] H.Daley, J.Perel, Phys.Rev.A 4 (1971) 162.
4] V.Aquilanti, G.Liuti, F.Vecchiocattivi, G G. .Volpi,
Entropie 42 (1971) 158.
Hemiquantal dynamical calculations have been
þ
carried out for the (CsNa ) collisional system for
energies below 1.60 keV in the LF. In order to
describe non-adiabatic effects as well as possible
the complete electronic wave function of the sys-
[5] V.Aquilanti, Faraday.Discuss.Chem.Soc.55 (1973) 231.
[
[
[
6] V.Aquilanti, G P. .Bellu, J.Chem.Phys.61 (1974) 1618.
7] V.Aquilanti, P.Casavecchia, J.Chem.Phys.64 (1976) 751.
8] V.Aquilanti, P.Casavecchia, G.Grossi, J.Chem.Phys.65
2
þ
tem has been expanded in a basis set of 18 R and
(
1976) 5518.
9] V.Aquilanti, P.Casavecchia, G.Grossi, J.Chem.Phys.68
2
1
2 P adiabatic states which include all experi-
[
mentally detected exit channels.Hemiquantal cal-
culations take into account both the radial and the
angular non adiabatic couplings between all basis
set electronic states and have been numerically
obtained from ab initio pseudopotential calcula-
tions for the pseudomonoelectronic quasimolecule
(1978) 1499.
10] V.Aquilanti, P.Casavecchia, G.Grossi, J.Chem.Phys.71
1979) 3547.
[
[
[
[
[
[
[
(
11] N.H. Tolk, C.W. White, S.H. Neff, W. Lichten, Phys. Rev.
Lett.11 (1973) 671.
12] N.Andersen, T.Andersen, K.Jensen, J.Phys.B 9 (1976)
1373.
13] J.Ø. Olsen, N. Andersen, T. Andersen, J. Phys. B 10 (1977)
þ
CsNa .
1
723.
14] B.Brunetti, F.Elisei, C.Vergari, Nuovo Cimento 63 (1981)
8.
Although in general a direct comparison of
measured emission cross sections values with those
obtained from hemiquantal calculations is not
possible, after some corrective considerations,
measured and calculated values are in relatively
good agreement considering the experimental
error, showing that hemiquantal dynamical cal-
culations are able to provide an adequate de-
scription of the non adiabatic process considered.
9
15] A.B €a hring, I.V. Hertel, E. Meyer, H. Schmidt, Phys. Rev.
Lett.53 (1984) 1433.
16] J.W. Thomsen, N. Andersen, E.E.B. Campbell, I.V. Hertel,
S.E. Nielsen, J. Phys. B 31 (1998) 3429.
[
[
17] V.Aquilanti, Z.Physik.Chem.90 (1974) 1.
18] C.F. Melius, W.A. Goddard III, Phys. Rev. Lett. 29 (1972)
9
75.
[19] W.L. McMillan, Phys. Rev. A 4 (1971) 69.
[20] R. Shingal, J.C. Noble, B.H. Brandsen, D.R. Flower,
J.Phys.B 19 (1986) 3951.
[
[
21] H.Nakamura, C.Zhu, J.Chem.Phys.101 (1994) 10630.
22] A.Aguilar, M.Albert ꢀı , J.de Andr ꢀe s, M.Gilibert, X.
Gim ꢀe nez, M.Gonz ꢀa lez, J.M. Lucas, M. Prieto, R. Say oꢀ s,
A.Sol ꢀe , Chem.Phys.Lett.220 (1994) 267.
Acknowledgements
This work has been supported by the Spanish
DGICYT project PB97-0919, the Spanish Minis-
try of Science and Technology project BQU2001-
3018 and the Generalitat de Catalunya-DURSI
project 2000SGR 00016.The Centre de Super-
[
[
[
23] J.de Andr ꢀe s, M.Prieto, T.Romero, JM. .Lucas, M.
Albert ꢀı , A.Aguilar, Chem.Phys.209 (1996) 338.
ꢀ
24] T.Romero, J.de Andr ꢀe s, M.Albert i, J.M. Lucas, A.
Aguilar, Chem.Phys.Lett.272 (1997) 271.
25] T.Romero, J.de Andr ꢀe s, J.Sogas, J M. .Bocanegra, M.
4
ꢀ
Alberti, J.M. Lucas, A. Aguilar, Chem. Phys. Lett. 281
computaci oꢀ de Catalunya C -CESCA-CEPBA is
(
1997) 74.
acknowledged for supplying the required CPU
time for dynamical calculations and implementa-
tion of the computer code.M S. .thanks to the
Universitat de Barcelona for a research grant.This
work has been done under auspicious of the COST
D23 ‘Metachem’ Action in Chemistry of the
European Commission.
ꢀ
[
26] T.Romero, J.de Andr ꢀe s, M.Albert i, J.M. Lucas, J.M.
Bocanegra, A.Aguilar, J.Sogas, Chem.Phys.Lett.292
(
27] A.Aguilar, J.de Andr ꢀe s, M.Albert i, J.M. Lucas, J. Sogas,
1998) 323.
ꢀ
[
J M. .Bocanegra, T.Romero, Phys.Chem.Chem.Phys.1
(
1999) 5607.
28] J.Sogas, M E. .Aricha, J.de Andr
ꢀ
ꢀe s, M.Albert i, J.M.
Lucas, A.Aguilar, Phys.Chem.Chem.Phys.3 (2001) 3638.
[
[
ꢀ
29] A.Aguilar, J.de Andr ꢀe s, T.Romero, M.Albert i, J.M.
Lucas, JM. .Bocanegra, J.Sogas, FX. .Gadea, in: R.
Campargue (Ed.), Atomic and Molecular Beams, The State
of the Art 2000, Springer, Berlin, 2001, p.599.
References
[
[
30] A.Salichs, Master Thesis, Universitat de Barcelona, 2000.
31] S.T. Chen, A.C. Gallagher, Phys. Rev. A 17 (1978) 571.
[
1] J. Perel, R.H. Vernon, H.L. Daley, Phys. Rev. A 138 (1965)
37.
9

J. de Andr ꢀe s et al. / Chemical Physics 281 (2002) 33–47
47
[
[
[
32] E.E. Nikitin, Theory of Chemical Elementary Processes in
Gases, Springer, Berlin, 1978.
[41] C.Zhu, H.Nakamura, J.Chem.Phys.102 (1995)
7448.
33] Ph. Durand, J.P. Malrieu, in: K.P. Lawley (Ed.), Adv.
Chem.Phys.LXVII (Part I) (1987) 321.
34] Ph.Durand, J C. .Berthelat, Theoret.Chim.Acta 55 (1980)
[42] J.C. Tully, in: W. Miller (Ed.), Dynamics of Molecular
Collisions Part B, Plenum Press, New York and London,
1976, p.220.
4
3;
Ph.Durand, J C. .Berthelat, M.Pelissier, Phys.Rev.A 21
1981) 1773.
[43] B.H. Bransden, Atomic Collision Theory, second ed., The
Benjamin/Cummings Publishing Company, London,
1983.
(
[
[
[
[
[
[
35] D.Pavolini, T.Gustavsson, F.Spielgelmann, J .P .Dauday,
J.Phys.B 22 (1989) 1721.
36] M.Foucrault, Ph.Millie, J P. .Daudey, J.Chem.Phys.96
[44] C.F. Gerad, P.O. Wheatley, Applied Numerical Analysis,
fifth ed., Addison-Wesley Publishing Company, New
York, 1970.
(
1992) 1257.
[45] L.L. Halcomb, D.J. Diestler, J. Chem. Phys. 84 (1986)
3130.
ꢀ
37] T.Romero, J.de Andr ꢀe s, M.Albert i, J.M. Lucas, J. Rubio,
J.P. Daudey, A. Aguilar, Chem. Phys. Lett. 261 (1996) 583.
38] J.Rubio, S.Zurita, J C. .Berthelat, F.Illas, Chem.Phys.
Lett.217 (1994) 283.
[46] M.E. Aricha, Master Thesis, Universitat de Barcelona,
1999.
[47] J. Reader, C.H. Corliss, in: D.R. Lide (Ed.), National
Bureau of Standards from the Handbook of Chemistry and
Physics, 77th ed., 1996–1997.
39] S.Magnier, Ph.Millie, O.Dulieu, F.Masnou-Seeuws, J.
Chem.Phys.98 (1993) 7113.
40] H.G. Jeung, J.P. Malrieu, J.P. Daudey, J. Chem. Phys. 77
[48] I.N. Levine, Molecular Spectroscopy, Wiley, New York,
1975.
(1982) 3571.